Uncertainty Propagation Calculator
How the uncertainties of two measured quantities combine into the uncertainty of a result — added in quadrature for a sum or difference, in relative terms for a product or quotient, with a power on the first quantity — against the pessimistic worst case.
Independent errors do not simply add: they combine as the root of the sum of squares, because they are as likely to cancel as to reinforce.
How the uncertainty propagation calculator works
Independent errors do not simply add: they combine as the root of the sum of squares, because they are as likely to cancel as to reinforce. For a sum or difference the absolute uncertainties combine that way; for a product or quotient it is the relative uncertainties, and a power multiplies its quantity's relative uncertainty by the exponent.
The worst case — adding the errors outright — is what you get if the errors are perfectly correlated, and it is the honest bound when you are not sure they are independent.
Formula: sum/difference: δz = √(δa² + δb²); product/quotient: δz/z = √((n δa/a)² + (δb/b)²)
Worked examples
| Inputs | Result | Note |
|---|---|---|
| A product | 10 ± 0.6 | 10 ± 0.64 |
| A difference | -1.5 ± 0.2 | −1.5 ± 0.22 |
| A squared quantity over another | 1.6 ± 0.1 | a counts twice |
FAQFrequently asked questions
Why add in quadrature?
Because independent random errors partly cancel. The variance of a sum is the sum of the variances, so the standard uncertainty is the root of the sum of squares.
When should I use the worst case?
When the errors might be correlated — the same instrument, the same systematic bias — or when a guaranteed bound matters more than a likely one.
What about a power?
A quantity raised to n has n times its relative uncertainty: measuring a radius to 2% gives an area to 4%. The power field applies that.
How many figures should I quote?
Round the uncertainty to one significant figure (two if it starts with 1) and the value to the same decimal place. The page does that in the quote row.
Is this the same as the GUM method?
It is the GUM's law of propagation of uncertainty for uncorrelated inputs, applied to the four basic operations. The general case uses partial derivatives and a covariance matrix.
Where these figures come from
- NIST — CODATA 2018 fundamental physical constants — G, g₀, R, c
- NIST Special Publication 811 — Guide for the use of the International System of Units — unit conversions
- The Engineering ToolBox — material properties — specific heats, expansion coefficients, densities
- National Physical Laboratory — the UK's national measurement institute
Last checked: September 2026. Constants are the CODATA 2018 values; formulas are the standard textbook forms.